7 PDES, FOURIER SERIES AND TRANSFORMS Last Updated: July 16, 2012

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1 Problem List 7.1 Fourier series expansion of absolute value function 7.2 Fourier series expansion of step function 7.3 Orthogonality of functions 7.4 Fourier transform of sinusoidal pulse 7.5 Introducing the Legendre polynomials 7.6 Using Fourier series to compute the infinite sum of reciprocal squares 7.7 Using complex Fourier series notation 7.8 Fourier series expansion of a periodic polynomial function 7.9 Finding the temperature field on a square plate 7.10 Basic properties of Fourier transforms 7.11 Fourier transform of triangularl pulse 7.12 Fourier transform of an important function from QM 7.13 You re an audio engineer! These problems are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Please share and/or modify. Back to Problem List 1

2 7.1 Fourier series expansion of absolute value function Given: Marino Fall 2011 For this problem, consider the periodic function f(x): (one period of the function is specified here) f(x) = x, for π 2 < x < 0 = x, for 0 < x < π 2 (a) (2 pt) Find the Fourier Series expansion for f(x). Be sure to include a sketch of the function. (b) (1 pt) To check that your solution is correct, use a computer, plot the sum of the first 3 non-zero terms in this series from π to π. It should resemble the sketch that you drew in part a. Back to Problem List 2

3 7.2 Fourier series expansion of step function Given: Marino Fall 2011 For this problem, consider the periodic function g(x): (one period of the function is specified here) g(x) = 1, for 1 < x < 1 = 0, for 1 < x < 5 (a) (2 pt) Find the Fourier Series expansion for g(x). Be sure to include a sketch of the function. (b) (1 pt) To check that your solution is correct, use a computer, plot the sum of the first first 4 non-zero terms in this series from -10 to 10. It should resemble the sketch that you drew in part a. Back to Problem List 3

4 7.3 Orthogonality of functions Given: Marino Fall 2011 Show that the functions x 2 and sin x are orthogonal on the interval (-1,1). (Hint: You should not need to work out the integral. What do you know about even and odd functions?) Back to Problem List 4

5 7.4 Fourier transform of sinusoidal pulse Given: Marino Fall 2011 Consider the following function: f(x) = sin x, for π 2 < x < π 2 = 0, for x > π 2 απ iα cos ( 2 (a) (1 pt) Demonstrate that the Fourier transform of f(x) is g(α) = ) π(1 α 2. Hint: It might ) help to express e iαx in terms of sines and cosines. Recalling the properties of integrals of odd and even functions can also save you some work. (b) (0.5 pt) Using your answer for g(α), express f(x) as a Fourier integral (i.e. substitute your result for g(α) into the Fourier integral for f(x)). Leave the answer as an integral. Do not evaluate the integral. Back to Problem List 5

6 7.5 Introducing the Legendre polynomials Given: Marino Fall 2011 We mentioned in class that the first few Legendre polynomials are P 0 (x) = 1 P 1 (x) = x P 2 (x) = 1 2 (3x2 1) (a) (0.5pt) Use the recursion relation to find P 3 (x). You must show all work to receive any credit. (b) (1 pt) Use the built-in Mathematica function LegendreP to find P 3 (x) and P 2 (x). Plot P 3 (x) and P 2 (x) from x = 1 to x = 1. (c) (0.5 pt) What kind of symmetry do P 3 (x) and P 2 (x) have? Are P 2 (x) and P 3 (x) orthogonal over the interval (-1,1)? Explain how you can tell that without having to evaluate any integrals. Back to Problem List 6

7 7.6 Using Fourier series to compute the infinite sum of reciprocal squares Given: Pollock Spring 2011, Spring 2012 Do you know the precise value of the summation of the reciprocals of the squares of the natural numbers, n=1 1 n? This is a famous problem in mathematical analysis with relevance to number theory. It was 2 first posed by Pietro Mengoli in 1644 and solved by Euler in Since the problem had withstood the attacks of the leading mathematicians of the day, Euler s solution brought him immediate fame when he was twenty-eight. Let s solve it using Fourier series (a tool that wasn t yet available to Euler!) (a) Define the periodic function f(x) = x 2 for π < x π and compute the Fourier series. (b) Evaluate the function at x = π and use the Fourier series to evaluate n=1 1 n. 2 The answer is. Your job is to show how this comes about!) n=1 1 n = π2 2 6 Back to Problem List 7

8 7.7 Using complex Fourier series notation Given: Pollock Spring 2011 Any periodic function, f(t) = f(t + T ) can also be Fourier expanded as with ω = 2π/T. f(t) = n= c n e iωnt (1) This is very similar to our usual expansion in terms of sin s and cos s, but combines them more elegantly using complex exponentials, and is the more common form used by physicists. Let s work out the details! (a) Show that 1 T/2 T T/2 ei(m n)tω dt = 0 if m n and 1 T/2 T T/2 ei(m n)tω dt = 1 if m = n. Here m, n are integers. (b) Use (a) to show that c m = 1 T/2 T T/2 e imtω f(t)dt, where m is just a dummy index. Hint! Look at the use of Fourier s trick from my lecture on Apr 7 (see concept test slides), or online lecture notes on p. 11a and 11b, or Boas Ch 7.7, and figure out how to apply it here. (c) Use Euler s formula e ix = cos x + i sin x to show that 1 2 (a n ib n ) n > 0 c n = a 0 n = (a n + ib n ) n < 0 Where the a s and b s are the usual Fourier coefficients we ve defined in class and Taylor. Note that, when expanding in complex exponentials like this, we win by having just one simple formula for the c n s (and integrals of exponentials are often easier). The price we pay is that we have to run our sum over negative n s Back to Problem List 8

9 7.8 Fourier series expansion of a periodic polynomial function Given: Pollock Spring 2011 For the periodic function f(t) = At for 1 < t 1 (with A a given constant) (a) Compute the Fourier coefficients a n and b n (b) Suppose this force drives a weakly damped oscillator with damping parameter β = 0.05 and a natural period T = 2. Find the long time motion x(t) of the oscillator. Discuss your answer. Use Mathematica to plot x(t) using the first four non-zero terms of the series, for 0 < t 10. (Set A=1 for this) (c) Repeat part b) for the same damping parameter, but with natural period T = 1. Briefly discuss any major differences between this result and what you had in the previous part. (d) For most of the possible natural frequencies the response of this particular driven oscillator is going to be weak. However, there are some particular natural frequencies at which you are going to see an enhanced response - determine which those are. (Be careful, think a bit about this!) Given: Pollock Spring 2012 For the periodic function f(t) = A(t 2 1/3) for 1 < t 1 (with A a given constant) (a) Compute Fourier coefficients a n and b n. integrals) (Some vanish - predict which ones before doing any (b) Suppose this force drives a weakly damped oscillator with damping parameter β = 0.05 and a natural period T = 2. Find the long time motion x(t) of the oscillator. Discuss your answer. Use Mathematica to plot x(t) using the first four non-zero terms of the series, for 0 < t 10. (Set A=1 for this) (c) Repeat part b) for the same damping parameter, but with natural period T = 1. Briefly discuss any major differences between this result and what you had in the previous part. (d) For most of the possible natural frequencies the response of this particular driven oscillator is going to be weak. However, there are some particular natural frequencies at which you are going to see an enhanced response - determine which those are. Back to Problem List 9

10 7.9 Finding the temperature field on a square plate Given: Pollock Spring 2011 For the rectangular plate problem shown above calculate the steady state temperature T (x, y) everywhere inside the plate if the boundary condition on the top surface is given by f(x) = { T0 0 < x a/2 0 a/2 < x a (2) For the situation above: (a) Code up your formula in Mathematica, adding up 30 terms in your sum to find T(x,y). For concreteness, please set a=1, b=2, and T 0 =100. Note that to define a function in Mathematica you need to be careful about syntax. If, for instance, T (x, y) = Σ n n sin(nπx)e nπy, the Mathematica syntax for that might be t[x,y ] := Sum[n Sin[n Pi x] Exp[-n Pi y],{n,1,30}] Watch out for the underscores for variables x and y on the left side (they do NOT appear again on the right side), and the colon before the equals sign. Don t capitalize functions that you invent, caps are reserved for MMA built-in functions. (b) As a check, set y=2, and just plot your T[x,2] (from x=0 to 1) to make sure you did that Fourier sum right. (That s the boundary condition we gave you, so you know what it should look like!) Comment on any interesting or surprising features you observe about this plot, is it exactly what you expected? (c) Plot your full T(x,y) using the function Plot3D. See the documentation center for the basic syntax of Plot3D. Briefly, discuss/document the key features of your plot. Please tell us explicitly what does the height of this graph represent, physically? Plot3D is very cool. You can rotate the resulting curve with your mouse to really get a good look at your result. Given: Pollock Spring 2012 You are a summer engineering-physics intern, working on a modified Rayleigh-Bernard convection ( experiment, and the edges of the rectangular bottom plate of the system are held at the temperatures as shown in the figure above. Back to Problem List 10

11 (a) Your advisor has asked you to determine the steady state temperature distribution T(x,y) for the plate, so another student can put that boundary information into a computational model for the convective flow she s constructed. The boundary condition on the top surface is given by { 0 0 < x a/2 f(x) = 2T 0 a/2 < x a (3) (b) You want to check that your solution makes sense. Code up your formula, adding up 30 terms in your sum to find T(x,y). For concreteness, please set a=1, b=2, and T 0 =100. The very first thing I would want to do to check this code would be to set y=2, and do a simple plot of T[x,2] (from x=0 to 1), to make sure that the Fourier sum is giving what you expect. (What DO you expect? Sketch it first, then plot it. Comment on any interesting or surprising features you observe about this plot, is it exactly what you expected? ) Some Mathematica reminders: if, for instance, T (x, y) = Σ n 2n sin(nπx)e nπy, the Mathematica syntax for that might be t[x,y ] := Sum[2n Sin[n Pi x] Exp[-n Pi y],{n,1,30}] Watch out for the underscores for variables x and y on the left side (they do NOT appear again on the right side), and the colon before the equals sign. Don t capitalize functions that you invent, caps are reserved for MMA built-in functions. (c) Now that you are more confident your (nasty!) Fourier work is probably ok, plot your full T(x,y) using a 3D plot function (MMA users will find Plot3D useful. See the documentation center for the basic syntax of Plot3D.) Briefly, discuss/document the key features of your plot. Please tell us explicitly what does the height of this graph represent, physically? Given these results, and the goal of your summer internship project, do you have any comments for your advisor? Plot3D is very cool. You can rotate the resulting curve with your mouse to really get a good look at your result. Extra credit, let s just play around with Fourier series a little! (i) For the rectangular plate problem shown above, calculate the steady state temperature T (x, y) everywhere inside the plate if the boundary condition on the top surface was f(x) = sin(3πx/a). (Note - this is much easier than the problem above, if you think about it! In fact, if you look back at what you did there, you shouldn t need to do any new calculations at all. ) (ii) Keep the boundary condition the same as part i on the bottom and left edges (T=0), and keep the boundary condition the same at the top edge too, i.e. T(x,y=b)= sin(3πx/a). But let s change the right edge: Let T(x=a,y) = sin(3πy/b). Find T(x,y) everywhere else. Hint: Once again, there is no need to do any real calculations here - you can pretty much just write down the answer if you think about it right! Back to Problem List 11

12 7.10 Basic properties of Fourier transforms Given: Pollock Spring 2011 (a) (2 pts) Show that if a function f(x) is even, then its Fourier transform g(α) (as defined in Boas Eq 12.2) is also an even function. (b) (2 pts) Show that in this case f(x) and g(α) can be written as Fourier cosine transforms and obtain Boas Eq (Assuming that f(x) is identical to Boas f c (x), how is g c (α) from Boas Eq related to g(α) of Boas Eq 12.2? Please note the typo in the top equation of Boas 12.15, the argument of g c should be α, not x. Back to Problem List 12

13 7.11 Fourier transform of triangular pulse Given: Pollock Spring 2011 For the function shown in the figure below (a) (1 pt) Qualitatively, how do you expect g(α), the Fourier transform of f(x), to change if you make a bigger or smaller? (b) (4 pts) Find the Fourier transform g(α). (c) (2 pts) Sketch g(α) (or plot it in Mathematica if you prefer). Was your prediction in part (a) correct? Explain. Figure 1: Back to Problem List 13

14 7.12 Fourier transform of an important function from QM Given: Pollock Spring 2011 The function e c x (with c a positive, known constant) is associated with bound states in quantum mechanics, and (pretty much as always in quantum mechanics) its Fourier transform is important to know. Find the Fourier transform of e c x, sketch it, and comment on any key features. As a mathematical aside, use your result to (easily!) compute 0 cos(αx) α 2 +1 dα. Back to Problem List 14

15 7.13 You re an audio engineer! Given: Pollock Spring 2012 You ve gotten a summer job interning as an audio engineer. Audio software produces periodic tones by summing pure frequencies (each with its own amplitude) - it s just summing up a Fourier series! Your boss has asked you to generate a tone with the following properties: It should have a fundamental frequency of 440 Hz (Middle A). On each cycle (i.e. 1/440th of a second), instead of being a pure cos or sin wave, your wave should start with a given amplitude A, ramp down linearly (rather than sinusoidally) to A in half a period, then ramp back up linearly back to +A, and repeat this pattern indefinitely. (a) Sketch the function you are trying to build (over a couple of periods). Clearly label your horizontal axis. Do you expect any of the terms (cosines or sines?) in the Fourier series to vanish? Briefly, why? (b) Compute the Fourier series for your waveform (which means coming up with a formula for the amplitude of each Fourier term.) (c) To check your solution, use a computer to plot the sum of the first 3 non-zero terms in your series (over a range of several periods.) It should resemble the sketch that you drew in part a. If you re using Mathematica, there is a function called Play, try it out! (Set A=1) Compare it to Play[Cos[440 * 2 Pi t], t,0,1] and describe in words how your tone is similar, and how it differs, from the pure tone. Back to Problem List 15